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The paper contains the proof of the estimate $$ \\mathbb E \\Psi(N_t) \\leq C_{\\Phi,\\Psi,X} \\mathbb E \\Phi(M_t),\\;\\;\\; t\\geq 0, $$ where $\\Phi, \\Psi:X \\to \\mathbb R_+$ are convex continuous functions and the least admissible constant $C_{\\Phi,\\Psi,X}$ coincides with the $\\Phi,\\Psi$-norm of the periodic Hilbert transform. As a corollary, it is shown that the $\\Phi,\\Psi$-norms of the periodic Hilbert transform, the Hilbert transform on the real l"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1805.03948","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2018-05-10T13:08:10Z","cross_cats_sorted":["math.CV","math.PR"],"title_canon_sha256":"0f12c6dc585b12a4dd5438b0eca477e1007f3c26672403c8b6e701dc90ae4a4d","abstract_canon_sha256":"6aa29a6563c3d5530a96e0ef4cc0bb5e16e9474a9f2ba3d9072df9593300e649"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:41:46.601030Z","signature_b64":"2wPe5aeSPBB7RRAzaX/GvNn/QSBwbhWJ4thoRHk6yg/zeJCotIjwx9EqYrM8RN0pZ8cANxDfYxjCP3pBS7VLCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"db9598592ed1088e40c638e2cc632cadb0634a8cf4dbc03904af505c6432ce6b","last_reissued_at":"2026-05-17T23:41:46.600370Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:41:46.600370Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The Hilbert transform and orthogonal martingales in Banach spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CV","math.PR"],"primary_cat":"math.FA","authors_text":"Adam Os\\k{e}kowski, Ivan Yaroslavtsev","submitted_at":"2018-05-10T13:08:10Z","abstract_excerpt":"Let $X$ be a given Banach space and let $M$, $N$ be two orthogonal $X$-valued local martingales such that $N$ is weakly differentially subordinate to $M$. 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